Spinorial Structure of $O(3)$ and Application to Dark Matter

Abstract

An $O(3)$ spinor, $\Phi$, as a doublet denoted by $2_D$ consists of an $SO(3)$ spinor, $\varphi$, and its complex conjugate, ${\varphi }^{\ast }$, which form $\Phi ={\left(\varphi ,{\varphi }^{\ast }\right)}^T$ to be identified with a Majorana-type spinor of $O(4)$. The four gamma matrices ${\Gamma }_{\mu }$ ($\mu =1\sim 4$) are given by ${\Gamma }_i=\text{diag.}\left({\tau }_i,{\tau }_i^{\ast }\right)$ ($i=1,2,3$) and ${\Gamma }_4=-{\tau }_2\otimes {\tau }_2$, where ${\tau }_i$ denote the Pauli matrices. The rotations and axis-reflections of $O(3)$ are, respectively, generated by ${\Sigma }_{ij}$ and ${\Sigma }_{i4}$, where ${\Sigma }_{\mu \nu }=[{\Gamma }_{\mu },{\Gamma }_{\nu }]/2i$. While $\Phi$ is regarded as a scalar, a fermionic $O(3)$ spinor is constructed out of an $SO(3)$ doublet Dirac spinor and its charge conjugate. These $O(3)$ spinors are restricted to be neutral and cannot carry the standard model quantum numbers because they contain particles and antiparticles. Our $O(3)$ spinors serve as candidates of dark matter. The $O(3)$ symmetry in particle physics is visible when the invariance of interactions is considered by explicitly including their complex conjugates. It is possible to introduce a dark gauge symmetry based on $SO(3)\times {\mathbf{Z}}_2$ equivalent to $O(3)$, where the ${\mathbf{Z}}_2$ parity is described by a $U(1)$ charge giving 1 for a particle and $-1$ for an antiparticle. The $SO(3)$ and $U(1)$ gauge bosons turn out to transform as the axial vector of $O(3)$ and the pseudoscalar of $O(3)$, respectively. This property is related to the consistent definition of the nonabelian field strength tensor of $O(3)$ or of the U(1) charge of the O(3)-transformed spinor. To see the feasibility of our dark matter models, we discuss scalar dark matter phenomenology based on the dark $U(1)$ gauge model.

31 pages, 8 figures; title slightly changed; dark matter phenomenology added; author added; accepted for publication in Nucl. Phys. B

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